Moy–Prasad filtration

The simplest example of a p-adic reductive group is ${\displaystyle \mathbb {Q} _{p}^{\times }}$, the multiplicative group of p-adic units. Since ${\displaystyle \mathbb {Q} _{p}^{\times }}$ is abelian, it has a unique parahoric subgroup, ${\displaystyle \mathbb {Z} _{p}^{\times }}$. The Moy–Prasad subgroups of ${\displaystyle \mathbb {Z} _{p}^{\times }}$ are the higher unit groups ${\displaystyle U^{(r)}}$, where for simplicity ${\displaystyle r}$ is a positive integer:

$${\displaystyle (\mathbb {Z} _{p}^{\times })_{r}=1+(p\,\mathbb {Z} _{p})^{r}=\{u\in \mathbb {Z} _{p}^{\times }:u\equiv 1{\bmod {p^{r}}}\}.}$$

The Lie algebra of ${\displaystyle \mathbb {Q} _{p}^{\times }}$ is ${\displaystyle \mathbb {Q} _{p}}$, and its Moy–Prasad subalgebras are the nonzero ideals of ${\displaystyle \mathbb {Z} _{p}}$:

$${\displaystyle (\mathbb {Z} _{p})_{r}=(p\,\mathbb {Z} _{p})^{r}=\{p^{r}a:a\in \mathbb {Z} _{p}\}.}$$

More generally, if ${\displaystyle r}$ is a positive real number then we use the floor function to define the ${\displaystyle r}$th Moy–Prasad subgroup and subalgebra:

$${\displaystyle (\mathbb {Z} _{p}^{\times })_{r}:=(\mathbb {Z} _{p}^{\times })_{\lfloor r\rfloor },\qquad (\mathbb {Z} _{p})_{r}:=(\mathbb {Z} _{p})_{\lfloor r\rfloor }}$$

This example illustrates the general phenomenon that although the Moy–Prasad filtration is indexed by the nonnegative real numbers, the filtration jumps only on a discrete, periodic subset, in this case, the natural numbers. In particular, it is usually the case that the ${\displaystyle r}$th and ${\displaystyle s}$th Moy–Prasad subgroups are equal if ${\displaystyle s}$ is only slightly larger than ${\displaystyle r}$.

Another important example of a p-adic reductive group is the general linear group ${\displaystyle {\text{GL}}_{n}(\mathbb {Q} _{p})}$; this example generalizes the previous one because ${\displaystyle {\text{GL}}_{1}(\mathbb {Q} _{p})=\mathbb {Q} _{p}^{\times }}$. Since ${\displaystyle {\text{GL}}_{n}(\mathbb {Q} _{p})}$ is nonabelian (when ${\displaystyle n\geq 2}$), it has infinitely many parahoric subgroups. One particular parahoric subgroup is ${\displaystyle {\text{GL}}_{n}(\mathbb {Z} _{p})}$. The Moy–Prasad subgroups of ${\displaystyle {\text{GL}}_{n}(\mathbb {Z} _{p})}$ are the subgroups of elements equal to the identity matrix ${\displaystyle 1}$ modulo high powers of ${\displaystyle p}$. Specifically, when ${\displaystyle r}$ is a positive integer we define

$${\displaystyle {\text{GL}}_{n}(\mathbb {Z} _{p})_{r}=1+(p\,{\text{M}}_{n}(\mathbb {Z} _{p}))^{r}=\{u\in {\text{M}}_{n}(\mathbb {Z} _{p}):u\equiv 1{\bmod {p^{r}}}\}.}$$

where ${\displaystyle {\text{M}}_{n}(\mathbb {Z} _{p})}$ is the algebra of n × n matrices with coefficients in ${\displaystyle \mathbb {Z} _{p}}$. The Lie algebra of ${\displaystyle {\text{GL}}_{n}(\mathbb {Q} _{p})}$ is ${\displaystyle {\text{M}}_{n}(\mathbb {Q} _{p})}$, and its Moy–Prasad subalgebras are the spaces of matrices equal to the zero matrix modulo high powers of ${\displaystyle p}$; when ${\displaystyle r}$ is a positive integer we define

$${\displaystyle {\text{M}}_{n}(\mathbb {Z} _{p})_{r}=(p\,{\text{M}}_{n}(\mathbb {Z} _{p}))^{r}=\{u\in {\text{M}}_{n}(\mathbb {Z} _{p}):u\equiv 0{\bmod {p^{r}}}\}.}$$

Finally, as before, if ${\displaystyle r}$ is a positive real number then we use the floor function to define the ${\displaystyle r}$th Moy–Prasad subgroup and subalgebra:

$${\displaystyle {\text{GL}}_{n}(\mathbb {Z} _{p})_{r}:={\text{GL}}_{n}(\mathbb {Z} _{p})_{\lfloor r\rfloor },\qquad {\text{M}}_{n}(\mathbb {Z} _{p})_{r}:={\text{M}}_{n}(\mathbb {Z} _{p})_{\lfloor r\rfloor }}$$

In this example, the Moy–Prasad groups would more commonly be denoted by ${\displaystyle {\text{GL}}_{n}(\mathbb {Q} _{p})_{x,r}}$ instead of ${\displaystyle {\text{GL}}_{n}(\mathbb {Z} _{p})_{r}}$, where ${\displaystyle x}$ is a point of the building of ${\displaystyle {\text{GL}}_{n}(\mathbb {Q} _{p})}$ whose corresponding parahoric subgroup is ${\displaystyle {\text{GL}}_{n}(\mathbb {Z} _{p}).}$

Since algebraic tori are a particular class of reductive groups, the theory of the Moy–Prasad filtration applies to them as well. It turns out, however, that the construction of the Moy–Prasad subgroups for a general reductive group relies on the construction for tori, so we begin by discussing the case where ${\displaystyle G=T}$ is a torus. Since the reduced building of a torus is a point there is only one choice for ${\displaystyle x}$, and so we will suppress ${\displaystyle x}$ from the notation and write ${\displaystyle T(k)_{r}:=T(k)_{x,r}}$.

First, consider the special case where ${\displaystyle T}$ is the Weil restriction of ${\displaystyle \mathbb {G} _{\text{m}}}$ along a finite separable extension ${\displaystyle \ell }$ of ${\displaystyle k}$, so that ${\displaystyle T(k)=\ell ^{\times }}$. In this case, we define ${\displaystyle T(k)_{r}}$ as the set of ${\displaystyle a\in \ell ^{\times }}$ such that ${\displaystyle {\text{val}}_{k}(x-1)\geq r}$, where ${\displaystyle {\text{val}}_{k}:\ell \to \mathbb {R} }$ is the unique extension of the valuation of ${\displaystyle k}$ to ${\displaystyle \ell }$.

A torus is said to be induced if it is the direct product of finitely many tori of the form considered in the previous paragraph. The ${\displaystyle r}$th Moy–Prasad subgroup of an induced torus is defined as the product of the ${\displaystyle r}$th Moy–Prasad subgroup of these factors.

Second, consider the case where ${\displaystyle r=0}$ but ${\displaystyle T}$ is an arbitrary torus. Here the Moy–Prasad subgroup ${\displaystyle T(k)_{0}}$ is defined as the integral points of the Néron lft-model of ${\displaystyle T}$.[8] This definition agrees with the previously given one when ${\displaystyle T}$ is an induced torus.

It turns out that every torus can be embedded in an induced torus. To define the Moy–Prasad subgroups of a general torus ${\displaystyle T}$, then, we choose an embedding of ${\displaystyle T}$ in an induced torus ${\displaystyle S}$ and define ${\displaystyle T(k)_{r}:=T(k)_{0}\cap S(k)_{r}}$. This construction is independent of the choice of induced torus and embedding.

For simplicity, we will first outline the construction of the Moy–Prasad subgroup ${\displaystyle G(k)_{x,r}}$ in the case where ${\displaystyle G}$ is split. After, we will comment on the general definition.

Let ${\displaystyle T}$ be a maximal split torus of ${\displaystyle G}$ whose apartment contains ${\displaystyle x}$, and let ${\displaystyle \Phi }$ be the root system of ${\displaystyle G}$ with respect to ${\displaystyle T}$.

For each ${\displaystyle \alpha \in \Phi }$, let ${\displaystyle U_{\alpha }}$ be the root subgroup of ${\displaystyle G}$ with respect to ${\displaystyle \alpha }$. As an abstract group ${\displaystyle U_{\alpha }}$ is isomorphic to ${\displaystyle \mathbb {G} _{\text{a}}}$, though there is no canonical isomorphism. The point ${\displaystyle x}$ determines, for each root ${\displaystyle \alpha }$, an additive valuation ${\displaystyle v_{\alpha ,x}:U_{\alpha }(k)\to \mathbb {R} }$. We define ${\displaystyle U_{\alpha }(k)_{x,r}:=\{u\in U_{\alpha }(k):v_{\alpha ,x}(u)\geq r\}}$.

Finally, the Moy–Prasad subgroup ${\displaystyle G(k)_{x,r}}$ is defined as the subgroup of ${\displaystyle G(k)}$ generated by the subgroups ${\displaystyle U_{\alpha }(k)_{x,r}}$ for ${\displaystyle \alpha \in \Phi }$ and the subgroup ${\displaystyle T(k)_{r}}$.

If ${\displaystyle G}$ is not split, then the Moy–Prasad subgroup ${\displaystyle G(k)_{x,r}}$ is defined by unramified descent from the quasi-split case, a standard trick in Bruhat–Tits theory. More specifically, one first generalizes the definition of the Moy–Prasad subgroups given above, which applies when ${\displaystyle G}$ is split, to the case where ${\displaystyle G}$ is only quasi-split, using the relative root system. From here, the Moy–Prasad subgroup can be defined for an arbitrary ${\displaystyle G}$ by passing to the maximal unramified extension ${\displaystyle k^{\text{nr}}}$ of ${\displaystyle k}$, a field over which every reductive group, and in particular ${\displaystyle G}$, is quasi-split, and then taking the fixed points of this Moy–Prasad group under the Galois group of ${\displaystyle k^{\text{nr}}}$ over ${\displaystyle k}$.

The ${\displaystyle k}$-group ${\displaystyle G}$ carries much more structure than the group ${\displaystyle G(k)}$ of rational points: the former is an algebraic variety whereas the second is only an abstract group. For this reason, there are many technical advantages to working not only with the abstract group ${\displaystyle G(k)}$, but also the variety ${\displaystyle G(k)}$. Similarly, although we described ${\displaystyle G(k)_{x,r}}$ as an abstract group, a certain subgroup of ${\displaystyle G(k)}$, it is desirable for ${\displaystyle G(k)_{x,r}}$ to be the group of integral points of a group scheme ${\displaystyle G_{x,r}}$ defined over the ring of integers, so that ${\displaystyle G(k)_{x,r}=G_{x,r}({\mathcal {O}}_{k})}$. In fact, it is possible to construct such a group scheme ${\displaystyle G_{x,r}}$.

Let ${\displaystyle {\mathfrak {g}}}$ be the Lie algebra of ${\displaystyle G}$. In a similar procedure as for reductive groups, namely, by defining Moy–Prasad filtrations on the Lie algebra of a torus and the Lie algebra of a root group, one can define the Moy–Prasad Lie algebras ${\displaystyle {\mathfrak {g}}_{x,r}}$ of ${\displaystyle {\mathfrak {g}}}$; they are free ${\displaystyle {\mathcal {O}}_{k}}$-modules, that is, ${\displaystyle {\mathcal {O}}_{k}}$-lattices in the ${\displaystyle k}$-vector space ${\displaystyle {\mathfrak {g}}}$. When ${\displaystyle r\geq 0}$, it turns out that ${\displaystyle {\mathfrak {g}}_{x,r}}$ is just the Lie algebra of the ${\displaystyle {\mathcal {O}}_{k}}$-group scheme ${\displaystyle G_{x,r}}$.

We have defined the Moy–Prasad filtration at the point ${\displaystyle x}$ to be indexed by the set ${\displaystyle \mathbb {R} }$ of real numbers. It is common in the subject to extend the indexing set slightly, to the set ${\displaystyle {\widetilde {\mathbb {R} }}}$ consisting of ${\displaystyle \mathbb {R} }$ and formal symbols ${\displaystyle r+}$ with ${\displaystyle r\in \mathbb {R} }$. The element ${\displaystyle r+}$ is thought of as being infinitesimally larger than ${\displaystyle r}$, and the filtration is extended to this case by defining ${\displaystyle G(k)_{x,r+}:=\bigcup _{s>r}G(k)_{x,s}}$. Since the valuation on ${\displaystyle k}$ is discrete, there is ${\displaystyle \varepsilon >0}$ such that ${\displaystyle G(k)_{x,r+}=G(k)_{x,r+\varepsilon }}$.